170 Solutions and Hints for Selected Exercises This implies that f is differentiable at aand f ′(a) =0. (c) We can verify that the function f (x) =|x| satisfies the requirement. From this problem, we see that it is only interesting to consider the class of functions that satisfy (4.8) when α ≤1. It is an exercise to show that the function f (x) =|x|1/2 satisfies this condition withℓ =1 and α=1/2. Exercise 4.3.3. Define the function h(x) =g(x)−f (x). Then h′(x) =g′(x)−f ′(x) ≥0 for all x ∈[x0,∞). Thus, his monotone increasing on this interval. It follows that h(x) ≥h(x0) =g(x0)−f (x0) =0 for all x ≥x0. Therefore, g(x) ≥f (x) for all x ≥x0. Exercise 4.3.5. Apply the mean value theorem twice. Exercise 4.3.6. Use proof by contradiction. SECTION 4.4 Exercise 4.4.5. Suppose that P(x) =a0 +a1x+· · ·+anx n. Then apply L’Hospital’s rule repeatedly. Exercise 4.4.6. We first consider the case where n=1 to get ideas for solving this problem in the general case. From the standard derivative theorems we get that the function is differentiable at any x̸ =0 with f ′(x) =2x−3e− 1 x2 = 2 x3 e− 1 x2 . Consider the limit lim x→0 f (x)−f (0) x−0 =lim x→0 e− 1 x2 x . Lettingt =1/x and applying L’Hospital rule yields lim x→0+ e− 1 x2 x =lim t→∞ t et2 =lim t→∞ 1 2tet2 =0. Similarly, lim x→0− e− 1 x2 x =0.
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